Finite Group Databasebeta

← #16855086

Subgroups — #16855086

Order 3968.

704 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14521884C211yesmaximal, characteristic
2#137×D31C212yesmaximal, complemented
3#14521619C211yesmaximal, characteristic, complemented
4#14521233C212yesmaximal, complemented
5#14521230C212yesmaximal, complemented
6#14521229C211yesmaximal, characteristic, complemented
7#14521227C211yesmaximal, characteristic, complemented
8#14521226C211yesmaximal, characteristic, complemented
9#14521064C212yesmaximal, complemented
10#14521030C211yesmaximal, characteristic, complemented
11C31×#455C211yesmaximal, characteristic, complemented
12C31×H(5,𝔽2)C2211yescharacteristic
13#11758567—22no—
14#11758567C2212yes—
15#41×D31—22no—
16#41×D31C2212yes—
17C2×Q8×D31C2211yescharacteristic, complemented
18#11758561—22no—
19#11758561—22no—
20#11758561C2211yescharacteristic
21C2×#92497—22no—
22C2×#92497—22no—
23C2×#92497C2211yescharacteristic
24C31×#137C2212yescomplemented
25C31×#124C2211yescharacteristic, complemented
26#11758528C2212yescomplemented
27#11758521C2211yescharacteristic, complemented
28#11758519C2212yescomplemented
29#11758513C2211yescharacteristic, complemented
30#11758495—22no—
31Q16×D31—22no—
32#11758492—22no—
33SD16×D31—22no—
34#11758486C2212yescomplemented
35M16×D31C2212yescomplemented
36#11758460—22no—
37#11758455—22no—
38#11758449C2211yescharacteristic, complemented
39#11758446C2211yescharacteristic, complemented
40C31×#104C2212yescomplemented
41C31×#100C2211yescharacteristic, complemented
42#11758425C2212yescomplemented
43#11758421C2211yescharacteristic, complemented
44#11758413C2212yescomplemented
45#11758412C2212yescomplemented
46#11758393C2212yescomplemented
47#41×C31—22no—
48#41×C31C2312yes—
49C2×Q8×C31C2311yescharacteristic
50C2×D4×C31—22no—
51C2×D4×C31—22no—
52C2×D4×C31C2311yescharacteristic
53C2×C22⋊D31—24no—
54C2×C22⋊D31—22no—
55C2×C22⋊D31—22no—
56C2×C22⋊D31D412yes—
57C22×Dic31—22no—
58C22×Dic31D412yes—
59#92499D412yes—
60Q8×D31—22no—
61Q8×D31—22no—
62Q8×D31—22no—
63Q8×D31D412yes—
64#92497—24no—
65#92497—24no—
66#92497—24no—
67#92497—22no—
68#92497—22no—
69#92497—24no—
70#92497—22no—
71#92497—22no—
72#92497—22no—
73#92497D412yes—
74D4×D31—24no—
75D4×D31—22no—
76D4×D31—22no—
77D4×D31D412yes—
78#92495—24no—
79#92495—22no—
80#92495—22no—
81#92495C2312yes—
82C2×C4×D31—22no—
83C2×C4×D31—22no—
84C2×C4×D31—21no—
85C2×C4×D31C2311yescharacteristic
86C2×Dic62—22no—
87C2×Dic62—22no—
88C2×Dic62—21no—
89C2×Dic62C2311yescharacteristic
90C2×Dic62C2311yescharacteristic
91Q16×C31—22no—
92SD16×C31—22no—
93M16×C31C2312yes—
94C22⋊C4×C31—22no—
95C42×C31C2311yescharacteristic
96C22⋊Dic31—22no—
97#92482—22no—
98#92482—22no—
99#92481—22no—
100#92480—22no—
101(C2×D31)⋊C4—21no—
102C4⋊Dic31—22no—
103Dic31⋊C4—21no—
104Dic31⋊C4—22no—
105C4×Dic31C2311yescharacteristic
106#92474C2312yes—
107Dic124—22no—
108#92470—22no—
109#92469—22no—
110C8×D31—22no—
111C23×C31—22no—
112C23×C31C2×D412yes—
113C22×D31—22no—
114C22×D31—22no—
115C22×D31C2×D411yescharacteristic
116Q8×C31—22no—
117Q8×C31C2×D412yes—
118D4×C31—24no—
119D4×C31—22no—
120D4×C31—22no—
121D4×C31C2×D412yes—
122C2×C4×C31—22no—
123C2×C4×C31—22no—
124C2×C4×C31—21no—
125C2×C4×C31—21no—
126C2×C4×C31C2411yescharacteristic
127C22⋊D31—24no—
128C22⋊D31—24no—
129C22⋊D31—22no—
130C22⋊D31—24no—
131C22⋊D31—22no—
132C22⋊D31—22no—
133C2×Dic31—24no—
134C2×Dic31—22no—
135C2×Dic31—22no—
136C2×Dic31—22no—
137C2×Dic31—22no—
138C2×Dic31—21no—
139C2×Dic31—21no—
140C2×Dic31C2×D412yes—
141C2×Dic31C2×D411yescharacteristic
142D124—22no—
143D124C2×D412yes—
144C4×D31—24no—
145C4×D31—22no—
146C4×D31—22no—
147C4×D31—22no—
148C4×D31—42no—
149C4×D31C2×D412yes—
150C4×D31C2×D412yes—
151Dic62—24no—
152Dic62—22no—
153Dic62—22no—
154Dic62—22no—
155Dic62—41no—
156Dic62—41no—
157Dic62C2×D412yes—
158C8×C31—22no—
159C31⋊C8—22no—
160#3022—311nomaximal, Sylow
161C22×C31—22no—
162C22×C31—22no—
163C22×C31—42no—
164C22×C31—82no—
165C22×C31C22×D411yescharacteristic
166C2×D31—22no—
167C2×D31—41no—
168C2×D31—82no—
169C2×D31—82no—
170C2×D31#12012yes—
171C4×C31—22no—
172C4×C31—41no—
173C4×C31—41no—
174C4×C31C22×D412yes—
175Dic31—22no—
176Dic31—22no—
177Dic31—41no—
178Dic31—41no—
179Dic31#12011yescharacteristic
180Dic31#12011yescharacteristic
181#585—311no—
182C2×#137—312no—
183#536—311no—
184#456—312no—
185#456—312no—
186#455—311no—
187#455—311no—
188#455—311no—
189#455D3111yescharacteristic, complemented
190#421—312no—
191#413—311no—
192C2×C31—21no—
193C2×C31—42no—
194C2×C31—42no—
195C2×C31C2×#12011yescharacteristic
196D31—22no—
197D31—42no—
1982−1+4—312no—
1992−1+4—622no—
200H(5,𝔽2)—311no—
201H(5,𝔽2)—622no—
202H(5,𝔽2)—622no—
203H(5,𝔽2)C2×D3111yescharacteristic
204C2×#41—311no—
205C2×#41—312no—
206C2×#41—622no—
207C2×#41—622no—
208C2×#41—622no—
209C22×Q8—311no—
210#137—312no—
211#137—312no—
212#137—312no—
213#137—622no—
214#137—622no—
215#137C2×D3112yescomplemented
216C2×Q16—622no—
217C2×SD16—622no—
218C2×M16—312no—
219C4⋊Q8—311no—
220C4⋊Q8—311no—
221#124—311no—
222#124C2×D3111yescharacteristic, complemented
223#123—622no—
224C22⋊Q8—622no—
225#117—311no—
226#104—312no—
227#104—312no—
228#104—312no—
229#104C2×D3112yescomplemented
230#101—312no—
231#100—311no—
232#100C2×D3111yescharacteristic, complemented
233C31#302211yescharacteristic, complemented, Sylow
234#41—22no—
235#41—312no—
236#41—312no—
237#41—312no—
238#41—622no—
239#41—622no—
240#41—624no—
241#41—622no—
242#41—622no—
243#41—622no—
244#41—624no—
245#41—622no—
246#41—624no—
247#41—624no—
248#41—624no—
249#41—622no—
250#41C22×D3112yes—
251C2×Q8—311no—
252C2×Q8—312no—
253C2×Q8—311no—
254C2×Q8—621no—
255C2×Q8—622no—
256C2×Q8—622no—
257C2×Q8—622no—
258C2×Q8—622no—
259C2×Q8—622no—
260C2×Q8C22×D3111yescharacteristic
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—312no—
264C2×D4—312no—
265C2×D4—622no—
266C2×D4—622no—
267C2×D4—624no—
268C2×D4—622no—
269C2×D4—624no—
270C2×D4—622no—
271C2×D4C22×D3111yescharacteristic
272C22×C4—311no—
273C22×C4—312no—
274C22×C4—622no—
275C22×C4—622no—
276C22×C4—621no—
277C22×C4—622no—
278Q16—22no—
279Q16—622no—
280Q16—622no—
281Q16—622no—
282SD16—22no—
283SD16—622no—
284SD16—622no—
285SD16—622no—
286M16—312no—
287M16—622no—
288M16C22×D3112yes—
289C2×C8—622no—
290C4⋊C4—621no—
291C4⋊C4—622no—
292C4⋊C4—622no—
293C22⋊C4—22no—
294C22⋊C4—621no—
295C22⋊C4—622no—
296C42—311no—
297C42C22×D3111yescharacteristic
298C23—22no—
299C23—311no—
300C23—622no—
301C23—622no—
302C23D4×D3112yes—
303Q8—22no—
304Q8—312no—
305Q8—622no—
306Q8—624no—
307Q8—622no—
308Q8—622no—
309Q8—1241no—
310Q8—1241no—
311Q8D4×D3112yes—
312D4—24no—
313D4—22no—
314D4—22no—
315D4—312no—
316D4—622no—
317D4—622no—
318D4—624no—
319D4—622no—
320D4—624no—
321D4—624no—
322D4—622no—
323D4D4×D3112yes—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—311no—
329C2×C4—312no—
330C2×C4—312no—
331C2×C4—312no—
332C2×C4—622no—
333C2×C4—622no—
334C2×C4—624no—
335C2×C4—622no—
336C2×C4—622no—
337C2×C4—622no—
338C2×C4—624no—
339C2×C4—622no—
340C2×C4—622no—
341C2×C4—621no—
342C2×C4—621no—
343C2×C4—1242no—
344C2×C4C23×D3111yescharacteristic
345C8—22no—
346C8—622no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—82no—
351C22—312no—
352C22—622no—
353C22—1241no—
354C22—2482no—
355C22—2482no—
356C22C2×D4×D3111yescharacteristic
357C4—22no—
358C4—41no—
359C4—41no—
360C4—311no—
361C4—311no—
362C4—622no—
363C4—622no—
364C4—1241no—
365C4—1241no—
366C4C2×D4×D3112yes—
367C2—21no—
368C2—42no—
369C2—42no—
370C2—622no—
371C2—1242no—
372C2#120×D3111yescharacteristic
subgroupG/H
centerC2#120×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#455C2
socleC2×C31C2×#120
prime radical#14521884C2
hypercenter#455D31
hypocenterC31#3022
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C2×C4 ◁ #455

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #137×D31C2semidirect
3. #14521619C2semidirect
4. #14521233C2semidirect
5. #14521230C2semidirect
6. #14521229C2semidirect
7. #14521227C2semidirect
8. #14521226C2semidirect
9. #14521064C2semidirect
10. #14521030C2semidirect
11. C31×#455C2semidirect
17. C2×Q8×D31C22semidirect
24. C31×#137C22semidirect
25. C31×#124C22semidirect
26. #11758528C22semidirect
27. #11758521C22semidirect
28. #11758519C22semidirect
29. #11758513C22semidirect
34. #11758486C22semidirect
35. M16×D31C22semidirect
38. #11758449C22semidirect
39. #11758446C22semidirect
40. C31×#104C22semidirect
41. C31×#100C22semidirect
42. #11758425C22semidirect
43. #11758421C22semidirect
44. #11758413C22semidirect
45. #11758412C22semidirect
46. #11758393C22semidirect
189. #455D31semidirect
215. #137C2×D31semidirect
222. #124C2×D31semidirect
229. #104C2×D31semidirect
232. #100C2×D31semidirect
233. C31#3022semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#145218842
#137×D312
#145216192
#145212332
#145212302
#145212292
#145212272
#145212262
#145210642
#145210302
C31×#4552
#302231